Population Genetics

Population genetics is a field of biology that studies the genetic
composition of biological populations, and the changes in genetic
composition that result from the operation of various factors,
including natural selection. Population geneticists pursue their goals
by developing abstract mathematical models of gene frequency dynamics,
trying to extract conclusions from those models about the likely
patterns of genetic variation in actual populations, and testing the
conclusions against empirical data. A number of the more robust
generalizations to emerge from population-genetic analysis are
discussed below.

Population genetics is intimately bound up with the study of evolution
and natural selection, and is often regarded as the theoretical
cornerstone of modern Darwinism. This is because natural selection is
one of the most important factors that can affect a population's
genetic composition. Natural selection occurs when some variants in a
population out-reproduce other variants as a result of being better
adapted to the environment, or ‘fitter’. Presuming the
fitness differences are at least partly due to genetic differences,
this will cause the population's genetic makeup to be altered over
time. By studying formal models of gene frequency change, population
geneticists therefore hope to shed light on the evolutionary process,
and to permit the consequences of different evolutionary hypotheses to
be explored in a quantitatively precise way.

The field of population genetics came into being in the 1920s and
1930s, thanks to the work of R.A. Fisher, J.B.S. Haldane and Sewall
Wright. Their achievement was to integrate the principles of Mendelian
genetics, which had been rediscovered at the turn of century, with
Darwinian natural selection. Though the compatibility of Darwinism with
Mendelian genetics is today taken for granted, in the early years of
the twentieth century it was not. Many of the early Mendelians did not
accept Darwin's ‘gradualist’ account of evolution,
believing instead that novel adaptations must arise in a single
mutational step; conversely, many of the early Darwinians did not
believe in Mendelian inheritance, often because of the erroneous belief
that it was incompatible with the process of evolutionary modification
as described by Darwin. By working out mathematically the consequences
of selection acting on a population obeying the Mendelian rules of
inheritance, Fisher, Haldane and Wright showed that Darwinism and
Mendelism were not just compatible but excellent bed fellows; this
played a key part in the formation of the ‘neo-Darwinian
synthesis’, and explains why population genetics came to occupy
so pivotal a role in evolutionary theory.

The discussion below is structured as follows. Section 1 outlines the
history of population genetics, focusing on major themes. Section 2
explains the Hardy-Weinberg principle, one of the most important
concepts in population genetics. Section 3 introduces the reader to
simple population-genetic models of the evolutionary process, and
discusses their significance. Section 4 discusses the status of
population genetics in modern biology, and some criticisms that have
been leveled against it. Section 5 examines some of the philosophical
issues raised by population genetics.

To understand how population genetics came into being, and to
appreciate its intellectual significance, a brief excursion into the
history of biology is necessary. Darwin's Origin of Species,
published in 1859, propounded two main theses: firstly, that modern
species were descended from common ancestors, and secondly that the
process of natural selection was the major mechanism of evolutionary
change. The first thesis quickly won acceptance in the scientific
community, but the second did not. Many people found it difficult to
accept that natural selection could play the explanatory role required
of it by Darwin's theory. This situation—accepting that
evolution had happened but doubting Darwin's account of what had
caused it to happen—persisted well into the twentieth
century (Bowler 1988).

Opposition to natural selection was understandable, for Darwin's
theory, though compelling, contained a major lacuna: an account of the
mechanism of inheritance. For evolution by natural selection to occur,
it is necessary that parents should tend to resemble their offspring;
otherwise, fitness-enhancing traits will have no tendency to spread
through a population. (For example, if fast zebras leave more offspring
then slow ones, this will only lead to evolutionary change if the
offspring of fast zebras are themselves fast runners.) In the
Origin, Darwin rested his argument on the observed fact that
offspring do tend to resemble their parents—‘the
strong principle of inheritance’—while admitting that he
did not know why this was. Darwin did later attempt an explicit theory
of inheritance, based on hypothetical entities called
‘gemmules’, but it turned out to have no basis in fact.

Darwin was deeply troubled by not having a proper understanding of the
inheritance mechanism, for it left him unable to rebut one of the most
powerful objections to his overall theory. For a population to evolve
by natural selection, the members of the population must vary—if
all organisms are identical, no selection can occur. So for selection
to gradually modify a population over a long period of time, in the
manner suggested by Darwin, a continual supply of variation is
needed. This was the basis for Fleeming Jenkins' famous objection to
Darwin, namely that the available variation would be used up too
fast. Jenkins' reasoning assumed a ‘blending’ theory of
inheritance, i.e. that an offspring's phenotypic traits are a
‘blend’ of those of its parents. (So for example, if a
short and a tall organism mate, the height of the offspring will be
intermediate between the two.) Jenkins argued that given blending
inheritance, a sexually reproducing population will become
phenotypically homogenous in just a few generations, far shorter than
the number of generations needed for natural selection to produce
complex adaptations.

Fortunately for Darwin's theory, inheritance does not actually work
the way Jenkins thought. The type of inheritance that we call
‘Mendelian’, after Gregor Mendel, is
‘particulate’ rather than
‘blending’—offspring inherit discrete hereditary
particles (genes) from their parents, which means that sexual
reproduction does not diminish the heritable variation present in the
population. (See section 2, ‘The Hardy-Weinberg
Principle’, below.) However, this realisation took a long time
to come, for two reasons. Firstly, Mendel's work was overlooked by the
scientific community for forty years. Secondly, even after the
rediscovery of Mendel's work at the turn of the twentieth century, it
was widely believed that Darwinian evolution and Mendelian inheritance
were incompatible. The early Mendelians did not accept that natural
selection played an important role in evolution, so were not well
placed to see that Mendel had given Darwin's theory the lifeline it
needed. The synthesis of Darwinism and Mendelism, which marked the
birth of modern population genetics, was achieved by a long and
tortuous route (Provine 1971).

The key ideas behind Mendel's theory of inheritance are
straightforward. In his experimental work on pea plants, Mendel
observed an unusual phenomenon. He began with two ‘pure
breeding’ lines, one producing plants with round seeds, the
other wrinkled seeds. He then crossed these to produce the first
daughter generation (the F1 generation). The F1 plants all had round
seeds—the wrinkled trait had disappeared from the
population. Mendel then crossed the F1 plants with each other to
produce the F2 generation. Strikingly, approximately one quarter of
the F2 plants had wrinkled seeds. So the wrinkled trait had made a
comeback, skipping a generation.

These and similar observations were explained by Mendel as follows.
He hypothesised that each plant contains a pair of
‘factors’ that together determine some aspect of its
phenotype—in this case, seed shape. A plant inherits one
factor from each of its parents. Suppose that there is one factor for
round seeds (R), another for wrinkled seeds (W). There are then three
possible types of plant: RR, RW and WW. An RR plant will have round
seeds, a WW plant wrinkled seeds. What about an RW plant? Mendel
suggested that it would have round seeds—the R factor is
‘dominant’ over the W factor. The observations could then
be easily explained. The initial pure-breeding lines were RR and WW.
The F1 plants were formed by RR × WW crosses, so were all of the
RW type and thus had round seeds. The F2 plants were formed by RW
× RW crosses, so contained a mixture of the RR, RW and WW
types. If we assume that each RW parent transmits the R and W factors
to its offspring with equal probability, then the F2 plants would
contain RR, RW and WW in the ratio 1:2:1. (This assumption is known as
Mendel's First Law or The Law of Segregation.) Since
RR and RW both have round seeds, this explains why three quarters of
the F2 plants had round seeds, one quarter wrinkled seeds.

Obviously, our modern understanding of heredity is vastly more
sophisticated than Mendel's, but the key elements of Mendel's
theory—discrete hereditary particles that come in different
types, dominance and recessiveness, and the law of
segregation—have turned out to be essentially correct. Mendel's
‘factors’ are the genes of modern population genetics, and
the alternative forms that a factor can take (e.g. R versus W in the
pea plant example) are known as the alleles of a gene. The
law of segregation is explained by the fact that during gametogenesis,
each gamete (sex cell) receives only one of each chromosome pair from
its parent organism. Other aspects of Mendel's theory have been
modified in the light of later discoveries. Mendel thought that most
phenotypic traits were controlled by a single pair of factors, like
seed shape in his pea plants, but it is now known that most traits are
affected by many pairs of genes, not just one. Mendel believed that
the pairs of factors responsible for different traits (e.g. seed shape
and flower colour) segregated independently of each other, but we now
know that this need not be so (see section 3.6, ‘Two-Locus
Models and Linkage’, below). Despite these points, Mendel's
theory marks a turning point in our understanding of inheritance.

The rediscovery of Mendel's work in 1900 did not lead the
scientific community to be converted to Mendelism overnight. The
dominant approach to the study of heredity at the time was biometry,
spearheaded by Karl Pearson in London, which involved statistical
analysis of the phenotypic variation found in natural populations.
Biometricians were mainly interested in continuously varying traits
such as height, rather than the ‘discrete’ traits such as
seed shape that Mendel studied, and were generally believers in
Darwinian gradualism. Opposed to the biometricians were the Mendelians,
spearheaded by William Bateson, who emphasized discontinuous variation,
and believed that major adaptive change could be produced by single
mutational steps, rather than by cumulative natural selection à
la Darwin. A heated controversy between the biometricians and the
Mendelians ensued. As a result, Mendelian inheritance came to be
associated with an anti-Darwinian view of evolution.

Population genetics as we know it today arose from the need to
reconcile Mendel with Darwin, a need which became increasingly urgent
as the empirical evidence for Mendelian inheritance began to pile up.
The first significant milestone was R.A. Fisher's 1918 paper,
‘The Correlation between Relatives on the Supposition of
Mendelian Inheritance’, which showed how the biometrical and
Mendelian research traditions could be unified. Fisher demonstrated
that if a given continuous trait, e.g. height, was affected by a large
number of Mendelian factors, each of which made a small difference to
the trait, then the trait would show an approximately normal
distribution in a population. Since the Darwinian process was widely
believed to work best on continuously varying traits, showing that the
distribution of such traits was compatible with Mendelism was an
important step towards reconciling Darwin with Mendel.

The full reconciliation was achieved in the 1920s and early 30s,
thanks to the mathematical work of Fisher, Haldane and Wright. Each of
these theorists developed formal models to explore how natural
selection, and other evolutionary forces such as mutation, would modify
the genetic composition of a Mendelian population over time. This work
marked a major step forward in our understanding of evolution, for it
enabled the consequences of various evolutionary hypotheses to be
explored quantitatively rather than just qualitatively. Verbal
arguments about what natural selection could or could not achieve, or
about the patterns of genetic variation to which it could give rise,
were replaced with explicit mathematical arguments. The strategy of
devising formal models to shed light on the process of evolution is
still the dominant research methodology in contemporary population
genetics.

There were important intellectual differences between Fisher,
Haldane and Wright, some of which have left legacies on the subsequent
development of the subject. One difference concerned their respective
attitudes towards natural selection. Fisher and Haldane were both
strong Darwinians—they believed that natural selection was by
far the most important factor affecting a population's genetic
composition. Wright did not downplay the role of natural selection, but
he believed that chance factors also played a crucial role in
evolution, as did migration between the constituent populations of a
species (See sections 3.3, ‘Random Drift’, and 3.4,
‘Migration’.) A related difference is that Wright emphasized
epistasis, or non-additive interactions between the genes within a
single genome, to a much greater extent than Fisher or Haldane. Despite
these differences, the work of all three was remarkably consonant in
overall approach.

The Hardy-Weinberg principle, discovered independently by G.H.
Hardy and W. Weinberg in 1908, is one of the simplest and most
important principles in population genetics. To illustrate the
principle, consider a large population of sexually reproducing
organisms. The organisms are assumed to be diploids, meaning
that they contain two copies of each chromosome, one received from each
parent. The gametes they produce are haploid, meaning that
they contain only one of each chromosome pair. During sexual fusion,
two haploid gametes fuse to form a diploid zygote, which then grows and
develops into an adult organism. Most multi-celled animals, and many
plants, have a lifecycle of this sort.

Suppose that at a given locus, or chromosomal ‘slot’,
there are two possible alleles, A1 and
A2; the locus is assumed to be on an autosome, not
a sex chromosome. With respect to the locus in question, there are
three possible genotypes in the population,
A1A1,
A1A2 and
A2A2 (just as in Mendel's pea
plant example above). Organisms with the
A1A1 and
A2A2 genotypes are called
homozygotes; those with the
A1A2 genotype are
heterozygotes. The proportions, or relative frequencies, of
the three genotypes in the overall population may be denoted
f(A1A1),
f(A1A2) and
f(A2A2) respectively,
where f(A1A1) +
f(A1A2) +
f(A2A2) = 1. It is
assumed that these genotypic frequencies are the same for both males
and females. The relative frequencies of the A and B
alleles in the population may be denoted p and q,
where p + q = 1.

The Hardy-Weinberg principle is about the relation between the allelic
and the genotypic frequencies. It states that if mating is random in
the population, and if the evolutionary forces of natural selection,
mutation, migration and drift are absent, then in the offspring
generation the genotypic and allelic frequencies will be related by
the following simple equations:

f(A1A1) =
p2,
f(A1A2) =
2pq,
f(A2A2) =
q2

Random mating means the absence of a genotypic correlation between
mating partners, i.e. the probability that a given organism mates with
an A1A1 partner, for example,
does not depend on the organism's own genotype, and similarly for the
probability of mating with a partner of one of the other two
types.

That random mating will lead the genotypes to be in the above
proportions (so-called Hardy-Weinberg proportions) is a
consequence of Mendel's law of segregation. To see this, note that
random mating is in effect equivalent to offspring being formed by
randomly picking pairs of gametes from a large ‘gamete
pool’ and fusing them into a zygote. The gamete pool contains
all the successful gametes of the parent organisms. Since we are
assuming the absence of selection, all parents contribute equal
numbers of gametes to the pool. By the law of segregation, an
A1A2 heterozygote produces
gametes bearing the A1 and A2
alleles in equal proportion. Therefore, the relative frequencies of
the A and B alleles in the gamete pool will be the
same as in the parental population, namely p and q
respectively. Given that the gamete pool is very large, when we pick
pairs of gametes from the pool at random, we will get the ordered
genotypic pairs {A1A1},
{A1A2},
{A2A1},
{A2A2} in the proportions
p2:pq:qp:q2.
But order does not matter, so we can regard the
{A1A2} and
{A2A1} pairs as equivalent,
giving the Hardy-Weinberg proportions for the unordered offspring
genotypes.

This simple derivation of the Hardy-Weinberg principle deals with two
alleles at a single locus, but can easily be extended to multiple
alleles. (Extension to more than one locus is trickier; see section
3.6, ‘Two-Locus Models and Linkage’, below.) For the
multi-allelic case, suppose there are n alleles at the locus,
A1 … An, with
relative frequencies of p1 …
pn respectively, where
p1 + p2 + … +
pn = 1. Assuming again that the
population is large, mating is random, evolutionary forces are absent,
and Mendel's law of segregation holds, then in the offspring
generation the frequency of the
AiAi
genotype will be pi2, and the
frequency of the (unordered)
AiAj
genotype (i ≠ j) will be
2pipj. It
is easy to see that the two allele case above is a special case of
this generalized principle.

Importantly, whatever the initial genotypic proportions, random mating
will automatically produce offspring in Hardy-Weinberg proportions
(for one-locus genotypes). So if generations are non-overlapping,
i.e. parents die as soon as they have reproduced, just one round of
random mating is needed to bring about Hardy-Weinberg proportions in
the whole population; if generations overlap, more than one round of
random mating is needed. Once Hardy-Weinberg proportions have been
achieved, they will be maintained in subsequent generations so long as
the population continues to mate at random and is unaffected by
evolutionary forces such as selection, mutation etc. The population is
then said to be in Hardy-Weinberg equilibrium—meaning
that the genotypic proportions are constant from generation to
generation.

The importance of the Hardy-Weinberg principle lies in the fact that it
contains the solution to the problem of blending that troubled Darwin.
As we saw, Jenkins argued that with sexual reproduction, the variation
in the population would be exhausted very rapidly. But the
Hardy-Weinberg principle teaches us that this is not so. Sexual
reproduction has no inherent tendency to destroy the genotypic
variation present in the population, for the genotypic proportions
remain constant over generations, given the assumptions noted above. It
is true that natural selection often tends to destroy
variation, and is thus a homogenizing force; but this is a quite
different matter. The ‘blending’ objection was that sexual
mixing itself would produce homogeneity, even in the absence
of selection, and the Hardy-Weinberg principle shows that this is
untrue.

Another benefit of the Hardy-Weinberg principle is that it greatly
simplifies the task of modelling evolutionary change. When a
population is in Hardy-Weinberg equilibrium, it is possible to track
the genotypic composition of the population by directly tracking the
allelic frequencies (or gametic frequencies). That this is so is
clear—for if we know the relative frequencies of all the alleles
(at a single locus), and know that the population is in Hardy-Weinberg
equilibrium, the entire genotype frequency distribution can be easily
computed. Were the population not in Hardy-Weinberg equilibrium, we
would need to explicitly track the genotype frequencies themselves,
which is more complicated.

Primarily for this reason, many population-genetic models assume that
Hardy-Weinberg equilibrium obtains; as we have seen, this is
tantamount to assuming that mating is random with respect to
genotype. But is this assumption empirically plausible? The answer is
sometimes but not always. In the human population, for example, mating
is close to random with respect to ABO blood group, so the genotype that
determines blood group is found in approximately Hardy-Weinberg proportions in many
populations (Hartl 1980). But mating is not random with respect to
height; on the contrary, people tend to choose mates similar in height
to themselves. So if we consider a genotype that influences height,
mating will not be random with respect to this genotype (see section
3.5 ‘Non-Random Mating’).

The geneticist W.J. Ewens has written of the Hardy-Weinberg principle,
‘it does not often happen that the most important theorem in any
subject is the easiest and most readily derived theorem for that
subject’ (1969, p. 1). The main importance of the principle, as
Ewens stresses, is not the gain in mathematical simplicity that it
permits, which is simply a beneficial side effect, but rather what it
teaches us about the preservation of genetic variation in a sexually
reproducing population.

Population geneticists usually define ‘evolution’ as any
change in a population's genetic composition over time. The four
factors that can bring about such a change are: natural selection,
mutation, random genetic drift, and migration into or out of the
population. (A fifth factor—changes to the mating
pattern—can change the genotype but not the
allele frequencies; many theorists would not count this as an
evolutionary change.) A brief introduction to the standard
population-genetic treatment of each of these factors is given
below.

Natural selection occurs when some genotypic variants in a population
enjoy a survival or reproduction advantage over others. The simplest
population-genetic model of natural selection assumes one autosomal
locus with two alleles, A1 and
A2, as above. The three diploid genotypes
A1A1,
A1A2 and
A2A2 have different fitnesses,
denoted by w11, w12 and
w22 respectively. These fitnesses are assumed to
be constant across generations. A genotype's fitness may be defined,
in this context, as the average number of successful gametes that an
organism of that genotype contributes to the next
generation—which depends on how well the organism survives, how
many matings it achieves, and how fertile it is. Unless
w11, w12 and
w22 are all equal, then natural selection will
occur, possibly leading the genetic composition of the population to
change.

Suppose that initially, i.e. before selection has operated, the zygote
genotypes are in Hardy-Weinberg proportions and the frequencies of the
A1 and A2 alleles are
p and q respectively, where p + q
= 1. The zygotes then grow to adulthood and reproduce, giving rise to
a new generation of offspring zygotes. Our task is to compute the
frequencies of A1 and A2 in
the second generation; let us denote these by p′ and
q′ respectively, where p′ +
q′ = 1. (Note that in both generations, we consider
gene frequencies at the zygotic stage; these may differ from the adult
gene frequencies if there is differential survivorship.)

In the first generation, the genotypic frequencies at the zygotic
stage are p2, 2pq and
q2 for
A1A1,
A1A2,
A2A2 respectively, by the
Hardy-Weinberg law. The three genotypes produce successful gametes in
proportion to their fitnesses, i.e. in the ratio
w11:w12:w22. The
average fitness in the population is
w =
p2w11 + 2pqw12 + q2w22, so the total number of successful gametes
produced is
Nw,
where N is
population size. Assuming there is no mutation, and that Mendel's law
of segregation holds, then an
A1A1 organism will produce
only A1 gametes, an
A2A2 organism will produce
only A2 gametes, and an
A1A2 organism will produce
A1 and A2 gametes in equal
proportion. Therefore, the proportion of A1
gametes, and thus the frequency of the A1 allele
in the second generation at the zygotic stage, is:

p′

=

[Np2w11 + ½ (N 2pqw12)] /
Nw

=

(p2w11 +
pqw12) /
w

(1)

Equation (1) is known as a ‘recurrence’ equation—it
expresses the frequency of the A1 allele in the second
generation in terms of its frequency in the first generation. The
change in frequency between generations can then be written as:

Δp

=

p′ − p

=

(p2w11 +
pqw12) /
w −
p

=

pq
[p(w11 −
w12) + q(w12
− w22)] /
w

(2)

If Δp > 0, then natural selection has led the
A1 allele to increase in frequency; if
Δp < 0 then selection has led the
A2 allele to increase in frequency. If
Δp = 0 then no gene frequency change has occurred,
i.e. the system is in allelic equilibrium. (Note, however, that the
condition Δp = 0 does not imply that no
natural selection has occurred; the condition for that is
w11 = w12 =
w22. It is possible for natural selection to
occur but to have no effect on gene frequencies.)

Equations (1) and (2) show, in precise terms, how fitness differences
between genotypes will lead to evolutionary change. This enables us to
explore the consequences of various different selective
regimes.

Suppose firstly that w11 >
w12 > w22, i.e. the
A1A1 homozygote is fitter than
the A1A2 heterozygote, which
in turn is fitter than the
A2A2 homozygote. By inspection
of equation (2), we can see that Δp must be positive
(so long as neither p nor q is zero, in which case
Δp = 0). So in each generation, the frequency of the
A1 allele will be greater than in the previous
generation, until it eventually reaches fixation and the
A2 allele is eliminated from the population. Once
the A1 allele reaches fixation, i.e. p =
1 and q = 0, no further evolutionary change will occur, for
if p = 1 then Δp = 0. This makes good sense
inutitively: since the A1 allele confers a fitness
advantage on organisms that carry it, its relative frequency in the
population will increase from generation to generation until it is
fixed.

It is obvious that analogous reasoning applies in the case where
w22 > w12 >
w11. Equation (2) tells us that Δp
must then be negative, so long as neither p nor q is
zero, so the A2 allele will sweep to fixation,
eliminating the A1 allele.

A more interesting
situation arises when the heterozygote is superior in fitness to both
of the homozygotes, i.e. w12 >
w11 and w12 >
w22—a phenomenon known as
heterosis. Intuitively it is clear what should happen in
this situation: an equilibrium situation should be reached in which
both alleles are present in the population. Equation (2) confirms this
intuition. It is easy to see that Δp = 0 if either
allele has gone to fixation (i.e. if p = 0 or q =
0), or, thirdly, if the following condition obtains:

p(w11 − w12) +
q(w12 − w22)
= 0

which reduces to

p = p* = (w12 −
w22) / (w12 −
w22) + (w12 −
w11)

(The asterisk indicates that this is an equilibrium condition.) Since
p must be non-negative, this condition can only be satisfied
if there is heterozygote superiority or heterozygote inferiority; it
represents an equilibrium state of the population in which both
alleles are present. This equilibrium is known as
polymorphic, by contrast with the monomorphic
equilibria that arise when either of the alleles has gone to
fixation. The possibility of polymorphic equilibrium is quite
significant. It teaches us that natural selection will not always
produce genetic homogeneity; in some cases, selection preserves the
genetic variation found in a population.

Numerous further questions about natural selection can be addressed
using a simple population-genetic model. For example, by incorporating
a parameter which measures the fitness differences between genotypes,
we can study the rate of evolutionary change, permitting us
to ask questions such as: how many generations are needed for
selection to increase the frequency of the A1
allele from 0.1 to 0.9? If a given deleterious allele is recessive,
how much longer will it take to eliminate it from the population than
if it were dominant? By permitting questions such as these to be
formulated and answered, population geneticists have brought
mathematical rigour to the theory of evolution, to an extent that
would have seemed unimaginable in Darwin's day.

Of course, the one-locus model discussed above is too simple to apply
to many real-life populations, for it incorporates simplifying
assumptions that are unlikely to hold true. Selection is rarely the
only evolutionary force in operation, genotypic fitnesses are unlikely
to be constant across generations, Mendelian segregation does not
always hold exactly, and so-on. Much research in population genetics
consists in devising more realistic evolutionary models, which rely on
fewer simplifying assumptions and are thus more complicated. But the
one-locus model illustrates the essence of population-genetic
reasoning, and the attendant clarification of the evolutionary process
that it brings.

Mutation is the ultimate source of genetic variation,
preventing populations from becoming genetically homogeneous in
situations where they otherwise would. Once mutation is taken into
account, the conclusions drawn in the previous section need to be
modified. Even if one allele is selectively superior to all others at
a given locus, it will not become fixed in the population; recurrent
mutation will ensure that other alleles are present at low frequency,
thus maintaining a degree of polymorphism. Population geneticists have
long been interested in exploring what happens when selection and
mutation act simultaneously.

Continuing with our one-locus, two allele model, let us suppose that
the A1 allele is selectively superior to
A2, but recurrent mutation from
A1 to A2 prevents
A1 from spreading to fixation. The rate of
mutation from A1 to A2 per
generation, i.e. the proportion of A1 alleles that
mutate every generation, is denoted u. (Empirical estimates
of mutation rates are typically in the region of 10-6.)
Back mutation from A2 to A1
can be ignored, because we are assuming that the
A2 allele is at a very low frequency in the
population, thanks to natural selection. What happens to the gene
frequency dynamics under these assumptions? Recall equation (1)
above, which expresses the frequency of the A1
allele in terms of its frequency in the previous generation. Since a
certain fraction (u) of the A1 alleles
will have mutated to A2, this recurrence equation
must be modified to:

p′ = (p2w11
+ pqw12)
(1 − u) /
w

to take account of mutation. As before, equilibrium is reached when
p′ = p, i.e. Δp = 0. The
condition for equilibrium is therefore:

p = p* = (p2w11 + pqw12) (1 − u) /
w

(3)

A useful simplification of equation (3) can be achieved by making some
assumptions about the genotype fitnesses, and adopting a new
notation. Let us suppose that the A2 allele is
completely recessive (as is often the case for deleterious
mutants). This means that the
A1A1 and
A1A2 genotypes have identical
fitness. Therefore, genotypic fitnesses can be written
w11 = 1, w12 = 1,
w22 = 1 − s, where s
denotes the difference in fitness of the
A2A2 homozygote from that of
the other two genotypes. (s is known as the selection
co-efficient against
A2A2). Since we are assuming
that the A2 allele is deleterious, it follows that
s > 0. Substituting these genotype fitnesses in equation
(3) yields:

p* = p (1 − u) / p2
+ 2pq + q2(1 − s)

which reduces to:

p* = 1 − (u/s)½

or equivalently (since p + q = 1):

q* =
(u/s)½

(4)

Equation (4) gives the equilibrium frequency of the
A2 allele, under the assumption that it is
completely recessive. Note that as u increases, q*
increases too. This is highly intuitive: the greater the mutation rate
from A1 to A2, the greater the
frequency of A2 that can be maintained at
equilibrium, for a given value of s. Conversely, as
s increases, q* decreases. This is also intuitive:
the stronger the selection against the
A2A2 homozygote, the lower the
equilibrium frequency of A2, for a given value of
u.

It is easy to see why equation (4) is said to describe
selection-mutation balance—natural selection is
continually removing A2 alleles from the
population, while mutation is continually re-creating them. Equation
(4) tells us the equilibrium frequency of A2 that
will be maintained, as a function of the rate of mutation from
A1 to A2 and the magnitude of
the selective disadvantage suffered by the
A2A2 homozygote. Importantly,
equation (4) was derived under the assumption that the
A2 allele is completely recessive, i.e. that the
A1A2 heterozygote is
phenotypically identical to the
A1A1 homozygote. However, it
is straightforward to derive similar equations for the cases where the
A2 allele is dominant, or partially dominant. If
A2 is dominant, or partially dominant, its
equilibrium frequency will be lower than if it is completely
recessive; for selection is more efficient at removing it from the
population. A deleterious allele that is recessive can
‘hide’ in heterozygotes, and thus escape the purging power
of selection, but a dominant allele cannot.

Before leaving this topic, one final point should be noted. Our
discussion has focused exclusively on deleterious mutations, i.e. ones
which reduce the fitness of their host organism. This may seem odd,
given that beneficial mutations play so crucial a role in the
evolutionary process. The reason is that in population genetics, a
major concern is to understand the causes of the genetic variability
found in biological populations. If a gene is beneficial, natural
selection is likely to be the major determinant of its equilibrium
frequency; the rate of sporadic mutation to that gene will play at
most a minor role. It is only where a gene is deleterious that
mutation plays a major role in maintaining it in a population.

Random genetic drift refers to the chance fluctuations in gene
frequency that arise in finite populations; it can be thought of as a
type of ‘sampling error’. In many evolutionary models, the
population is assumed to be infinite, or very large, precisely in
order to abstract away from chance fluctuations. But though
mathematically convenient, this assumption is often unrealistic. In
real life, chance factors will invariably play a role, particularly in
small populations. The term 'random drift' is sometimes used in broad
sense, to refer to any stochastic factors that affect gene frequencies
in a population, including for example chance fluctuations in survival
and mating success; and sometimes in a narrower sense, to refer to the
random sampling of gametes to form the offspring generation (which
arises because organisms produce many more gametes than will ever make
it into a fertilized zygote). The broader sense is used here.

To understand the nature of random drift, consider a simple example.
A population contains just ten organisms, five of type A and
five of type B; the organisms reproduce asexually and beget
offspring of the same type. Suppose that neither type is selectively
superior to the other—both are equally well-adapted to the
environment. However, this does not imply that the two types will
produce identical numbers of offspring, for chance factors may play a
role. For example, it is possible that all the type Bs might
die by accident before reproducing; in which case the frequency of
B in the second generation will fall to zero. If so, then the
decline of the B type (and thus the spread of the A
type) is the result of random drift. Evolutionists are often
interested in knowing whether a given gene frequency change is the
result of drift, selection, or some combination of the two.

The label ‘random drift’ is slightly misleading. In saying
that the spread of the A type is due to random drift, or
chance, we do not mean that no cause can be found of its spread. In
theory, we could presumably discover the complete causal story about
why each organism in the population left exactly the number of
offspring that it did. In ascribing the evolutionary change to random
drift, we are not denying that there is such a causal story to be
told. Rather, we mean that the spread of the A type was not
due to its adaptive superiority over the B type. Put
differently, the A and the B types had the same
expected number of offspring, so were equally fit; but the
A types had a greater actual number of offspring. In
a finite population, actual reproductive output will almost always
deviate from expectation, leading to evolutionary change.

An analogy with coin tossing can illuminate random drift. Suppose a
fair coin is tossed ten times. The probability of heads on any one
toss is ½, and so the expected frequency of heads in
the sequence of ten is 50%. But the probability of actually
getting half heads and half tails is only 242/1024, or approximately
23.6%. So even though the coin is fair, we are unlikely to get equal
proportions of heads and tails in a sequence of ten tosses; some
deviation from expectation is more probable than not. In just the same
way, even though the A and B types are equally fit
in the example above, it is likely that some evolutionary change will
occur. This analogy also illustrates the role of population size. If
we tossed the coin a hundred times rather than ten, the proportion of
heads would probably be very close to ½. In just the same way,
the larger the population, the less important the effect of random
drift on gene frequencies; in the infinite limit, drift has no
effect.

Drift greatly complicates the task facing the population geneticist.
In the example above, it is obviously impossible to deduce
the composition of the population in the second generation from its
composition in the first generation; at most, we can hope to deduce
the probability distribution over all the possible compositions. So
once drift is taken into account, no simple recurrence relation for
gene frequencies, of the sort expressed in equation (1) above, can be
derived. To analyse the evolutionary consequences of drift, population
geneticists use a mathematical technique known as diffusion modelling,
which is beyond the scope of this article; see Gillespie (2004) or
Rice (2004) for good introductions. However, many of these
consequences are quite intuitive, and can be understood without the
mathematics.

One important effect of random drift is to decrease the degree of
heterozygosity in a population over time. This happens because, given
enough time, any finite population will eventually become homozygous
through drift (though if the population is large, the approach to
homozygosity will be slow.) It is easy to see why this is—for
gene frequencies of 0 and 1 are ‘absorbing boundaries’,
meaning that once the boundary is reached, there is no way back from
it (apart from mutation). So eventually, a given allele will
eventually become fixed in a population, or go extinct, the latter
being the more likely fate. Indeed mathematical models show that a
neutral allele arising by mutation has a very low probability of
becoming fixed in a population; the larger the population, the lower
the probability of fixation.

Another important effect of random drift is to cause the different
subpopulations of a species to diverge genetically from each other,
as the chance accumulation of alleles will probably proceed
differently in each, particularly if the alleles confer little
selective advantage or disadvantage. By chance, one population may
become fixed for allele A1, while a second
population becomes fixed for another allele
A2. This possibility is an important one, for if
we ignore it, we may mistakenly conclude that the
A1 allele must have been advantageous in the
environment of the first population, the A2 allele
in the environment of the second, i.e. that selection was responsible
for the genetic differentiation. Such an explanation might be
right, but it is not the only one—random drift provides an
alternative.

The question of whether drift or selection plays a more important role
in molecular evolution was much debated in the 1960s and 1970s; it lay
at the heart of the heated controversy between
‘selectionists’ and ‘neutralists’ (see
Dietrich 1994). The neutralist camp, headed by M. Kimura, argued that
most molecular variants had no effect on phenotype, so were not
subject to natural selection; random drift was the major determinant
of their fate. Kimura argued that the apparently constant rate at
which the amino acid sequences of proteins evolved, and the extent of
genetic polymorphism observed in natural populations, could best be
explained by the neutralist hypothesis (Kimura 1977,
1994). Selectionists countered that natural selection was also capable
of explaining the molecular data. In recent years, the controversy has
subsided somewhat, without a clear victory for either side. Most
biologists believe that some molecular variants are indeed neutral,
though fewer than were claimed by the original neutralists.

Migration into or out of a population is the fourth and final factor
that can affect its genetic composition. Obviously, if immigrants are
genetically different from the population they are entering, this will
cause the population's genetic composition to be altered. The
evolutionary importance of migration stems from the fact that many
species are composed of a number of distinct subpopulations, largely
isolated from each other but connected by occasional migration. (For
an extreme example of population subdivision, think of ant colonies.)
Migration between subpopulations gives rise to gene flow, which acts
as a sort of ‘glue’, limiting the extent to which
subpopulations can diverge from each other genetically.

The simplest model for analysing migration assumes that a given
population receives a number of migrants each generation, but sends
out no emigrants. Suppose the frequency of the A1
allele in the resident population is p, and the frequency of
the A1 allele among the migrants arriving in the
population is pm. The proportion of
migrants coming into the population each generation is m
(i.e. as a proportion of the resident population.) So post-migration,
the frequency of the A1 allele in the population
is:

p′ = (1 − m) p +
mpm

The change in gene frequency across generations is therefore:

Δp

=

p′ − p

=

− m (p −
pm)

Therefore, migration will increase the frequency of the
A1 allele if pm >
p, decrease its frequency if p >
pm, and leave its frequency unchanged if
p = pm. It is then a
straightforward matter to derive an equation giving the gene frequency
in generation t as a function of its initial frequency and
the rate of migration. The equation is:

pt = pm +
(p0 − pm)(1
− m)t

where p0 is the initial frequency of the
A1 allele in the population, i.e. before any
migration has taken place. Since the expression (1 −
m)t tends towards zero as t
grows large, it is easy to see that equilibrium is reached when
pt = pm,
i.e. when the gene frequency of the migrants equals the gene frequency
of the resident population.

This simple model assumes that migration is the only factor affecting
gene frequency at the locus, but this is unlikely to be the case. So
it is necessary to consider how migration will interact with
selection, drift and mutation. A balance between migration and
selection can lead to the maintenance of a deleterious allele in a
population, in a manner closely analogous to mutation-selection
balance, discussed above. The interaction between migration and drift
is especially interesting. We have seen that drift can lead the
separate subpopulations of a species to diverge
genetically. Migration opposes this trend—it is a homogenising
force that tends to make subpopulations more alike. Mathematical
models suggest that that even a fairly small rate of migration will be
sufficient to prevent the subpopulations of a species from diverging
genetically. Some theorists have used this to argue against the
evolutionary importance of group selection, on the grounds that
genetic differences between groups, which are essential for group
selection to operate, are unlikely to persist in the face of
migration.

Recall that the Hardy-Weinberg law, the starting point for most
population-genetic analysis, was derived under the assumption of
random mating. But departures from random mating are actually quite
common. Organisms may tend to choose mates who are similar to them
phenotypically or genotypically—a mating system known as
‘positive assortment’. Alternatively, organisms may choose
mates dissimilar to them—‘negative assortment’.
Another type of departure from random mating is inbreeding, or
preferentially mating with relatives.

Analysing the consequences of non-random mating is quite complicated,
but some conclusions are fairly easily seen. Firstly and most
importantly, non-random mating does not in itself affect gene
frequencies (so is not an evolutionary ‘force’ on a par
with selection, mutation, migration and drift); rather, it affects
genotype frequencies. To appreciate this point, note that the gene
frequency of a population, at the zygotic stage, is equal to the gene
frequency in the pool of successful gametes from which the zygotes are
formed. The pattern of mating simply determines the way in which
haploid gametes are ‘packaged’ into diploid zygotes. Thus
if a random mating population suddenly starts to mate non-randomly,
this will have no effect on gene frequencies.

Secondly, positive assortative mating will tend to decrease the
proportion of heterozygotes in the population, thus increasing the
genotypic variance. To see this, consider again a single locus with
two alleles, A1 and A2, with
frequencies p and q in a given population. Initially
the population is at Hardy-Weinberg equilibrium, so the proportion of
A1A2 heterozygotes is
2pq. If the population then starts to mate
completely assortatively, i.e. mating only occurs between organisms of
identical genotype, it is obvious that the proportion of heterozygotes
must decline. For A1A1 ×
A1A1 and
A2A2 ×
A2A2 matings will produce no
heterozygotes; and only half the progeny of
A1A2 ×
A1A2 matings will be
heterozygotic. So the proportion of heterozygotes in the second
generation must be less than 2pq. Conversely,
negative assortment will tend to increase the proportion of
heterozygotes from what it would be under Hardy-Weinberg
equilibrium.

What about inbreeding? In general, inbreeding will tend to increase
the homozygosity of a population, like positive assortment. The reason
for this is obvious—relatives tend to be more genotypically
similar than randomly chosen members of the population. In the
majority of species, including the human species, inbreeding has
negative effects on organismic fitness—a phenomenon known as
‘inbreeding depression’. The explanation for this is that
deleterious alleles often tend to be recessive, so have no phenotypic
effect when found in heterozygotes. Inbreeding reduces the proportion
of heterozygotes, making recessive alleles more likely to be found in
homozygotes where their negative phenotypic effects become
apparent. The converse phenomenon—‘hybrid vigour’
resulting from outbreeding—is widely utilised by animal and
plant breeders.

So far, our exposition has dealt with gene frequency change at a
single locus, which is the simplest sort of population-genetic
analysis. However, in practice it is unlikely that an organism's
fitness will depend on its single-locus genotype, so there is a limit
on the extent to which single-locus models can illuminate the
evolutionary process. The aim of two-locus (and more generally,
multi-locus) models is to track changes in gene frequency at more than
one locus simultaneously. Such models are invariably more complicated
that their single-locus counterparts, but achieve greater realism.

The simplest two-locus model assumes two autosomal loci, A
and B, each with two alleles, A1 and
A2, B1 and
B2 respectively. Thus there are four types of
haploid gamete in the
population—A1B1,
A1B2,
A2B1 and
A2B2—whose frequencies
we will denote by x1, x2,
x3 and x4 respectively. (Note
that the xi are not allele
frequencies; in the two-locus case, we cannot equate ‘gamete
frequency’ with ‘allelic frequency’, as is possible
for a single locus.) Diploid organisms are formed by the fusion of two
gametes, as before. Thus there are ten possible diploid genotypes in
the population—found by taking each gamete type in combination
with every other.

In the one-locus case, we saw that in a large randomly mating
population, there is a simple relationship between the frequencies of
the gamete types and of the zygotic genotypes that they form. In the
two-locus case, the same relationship holds. Thus for example, the
frequency of the A1B1 /
A1B1 genotype will be
(x1)2; the frequency of the
A1B1 /
A2B1 genotype will be
2x1x3, and so-on. (This can be
established rigorously with an argument based on random sampling of
gametes, analogous to the argument used in the one-locus case.) The
first aspect of the Hardy-Weinberg law—genotypic frequencies
given by the square of the array of gametic
frequencies—therefore transposes neatly to the two-locus
case. However, the second aspect of the Hardy-Weinberg
law—stable genotypic frequencies after one round of random
mating—does not generally apply in the two-locus case, as we
will see.

A key concept in two-locus population genetics is that of
linkage, or lack of independence between the two loci. To
understand linkage, consider the set of gametes produced by an
organism of the A1B1 /
A2B2 genotype, i.e. a double
heterozygote. If the two loci are unlinked, then the
composition of this set will be {¼
A1B1, ¼
A1B2, ¼
A2B1, ¼
A2B2}, i.e. all four gamete
types will be equally represented. (We are presuming that Mendel's
first law holds at both loci.) So unlinked loci are
independent—which allele a gamete has at the A locus
tells us nothing about which allele it has at the B
locus. The opposite extreme is perfect linkage. If the two loci are
perfectly linked, then the set of gametes produced by the
A1B1 /
A2B2 double heterozygote has
the composition {½
A1B1, ½
A2B2}; this means that if a
gamete receives the A1 allele at the A
locus, it necessarily receives the B1 allele at
the B locus and vice versa.

In physical terms, perfect linkage means that the A and
B loci are located close together on the same chromosome; the
alleles at the two loci are thus inherited as a single unit. Unlinked
loci are either on different chromosomes, or on the same chromosome
but separated by a considerable distance, hence likely to be broken up
by recombination. Where the loci are on the same chromosome, perfect
linkage and complete lack of linkage are two ends of a continuum. The
degree of linkage is measured by the recombination fractionr, where 0 ≤ r ≤ ½. The composition of
the set of gametes produced by an organism of the
A1B1 /
A2B2 genotype can be written
in terms of r, as follows:

A1B1

½ (1 − r)

A1B2

½ r

A2B1

½ r

A2B2

½ (1 − r)

It is easy to see that r = ½ means that the loci are
unlinked, so all four gamete types are produced in equal proportion,
while r = 0 means that they are perfectly linked.

In a two-locus model, the gametic (and therefore genotypic)
frequencies need not be constant across generations, even in the
absence of selection, mutation, migration and drift, unlike in the
one-locus case. (Though allelic frequencies will of course
be constant, in the absence of any evolutionary forces.) It is
possible to write recurrence equations for the gamete frequencies, as
a function of their frequencies in the previous generation plus the
recombination fraction. The equations are:

(See Ewens 1969 or Edwards 2000 for an explicit derivation of these
equations.)

From the recurrence equations, it is easy to see that gametic (and
thus genotypic) frequencies will be stable across generations, i.e.
xi′ =
xi for each i, under either of
two conditions: (i) r = 0, or (ii)
x2x3 −
x1x4 = 0. Condition (i) means
that the two loci are perfectly linked, and thus in effect behaving as
one locus; condition (ii) means that the two loci are in
‘linkage equilibrium’, which means that the alleles at the
A-locus are in random association with the alleles at the
B-locus. More precisely, linkage equilibrium means that the
population-wide frequency of the
AiBi gamete
is equal to the frequency of the Ai
allele multiplied by the frequency of the
Bi allele.

An important result in two-locus theory shows that, given random
mating, the quantity (x2x3
− x1x4) will decrease
every generation until it reaches zero—at which point the
genotype frequencies will be in equilibrium. So a population initially
in linkage disequilibrium will approach linkage equilibrium over a
number of generations. The rate of approach depends on the value of
r, the recombination fraction. Note the contrast with the
one-locus case, where just one round of random mating is sufficient to
bring the genotype frequencies into equilibrium.

The basic models of classical population genetics, expounded in the
previous sections, have been around for nearly a century; they derive
from the work of Fisher, Haldane and Wright in the 1920s. Modern
population genetics has built on this theoretical edifice in a number
of ways, most notably by integrating the theory with data from
molecular biology. Advances in molecular biology have produced an
enormous supply of data on the genetic variability of actual
populations, which has enabled a link to be forged between abstract
population-genetic models and empirical data. This is not in itself a
new development: the selectionist-neutralist controversy of the 1960s,
mentioned above, was fuelled by the then-new data on protein
polymorphism in fruit-fly populations (see Lewontin and Hubby
1966). More recently, extensive data sets on variation at the DNA
rather than the protein level have become available; this has led to
the rise of ‘molecular population genetics’ and an
associated set of ideas known as ‘coalescent theory’ (see
Wakeley 2006). Unlike traditional population-genetic analysis, which
tries to determine how a given population will evolve in the future,
coalescent theory tries to reconstruct the ancestral state of a
population from its current state, based on the idea that all the
genes in a population ultimately derive from a single common
ancestor. Coalescent theory underpins much contemporary research in
population genetics.

The status of population genetics in contemporary biology is an
interesting issue. Despite its centrality to evolutionary theory, and
its historical importance, population genetics is not without its
critics. Some argue that population geneticists have devoted too much
energy to developing theoretical models, often with great mathematical
ingenuity, and too little to actually testing the models against
empirical data (Wade 2005). Others argue that population-genetic
models are usually too idealized to shed any real light on the
evolutionary process, and are limited in what they can teach us about
phenotypic evolution (Pigliucci 2008). Still others have argued that,
historically, population genetics has had a relatively minor impact on
the actual practice of most evolutionary biologists, despite the
lip-service often paid to it (Lewontin 1980). However, not all
biologists accept these criticisms. Thus the geneticist Michael Lynch
(2007), for example, has recently written that “nothing in
biology makes sense except in the light of population genetics”,
in a reference to Dobzhanksy's famous dictum; see Bromham (2009) and
Pigliucci (2008) for discussion.

Population-genetic models of evolution have also been criticised on
the grounds that few phenotypic traits are controlled by genotype at a
single locus, or even two or three loci. (Multi-locus
population-genetic models do exist, but they tend to be extremely
complicated.) There is an alternative body of theory, known as
quantitative genetics, which deals with so-called 'polygenic' or
'continuous' traits, such as height, which are thought to be affected
by genes at many different loci in the genome, rather than just one or
two; see Falconer (1995) for a good introduction. Quantitative
genetics employs a quite different methodology from population
genetics. The latter, as we have seen, aims to track gene and genotype
frequencies across generations. By contrast, quantitative genetics
does not directly deal with gene frequencies; the aim is to track the
phenotype distribution, or moments of the distribution such as the
mean or the variance, across generations. Though widely used by animal
and plant breeders, quantitative genetics is usually regarded as a
less fundamental body of theory than population genetics, given its
'phenotypic' orientation, and plays less of a role in evolutionary
theorising. Nonetheless, the relationship between population and
quantitative genetics is essentially harmonious.

A different criticism of the population-genetic approach to evolution
is that it ignores embryological development; this criticism really
applies to the evolutionary theory of the ‘modern
synthesis’ era more generally, which had population genetics at
its core. As we have seen, population-genetic reasoning assumes that
an organism's genes somehow affect its phenotype, and thus its
fitness, but it is silent about the details of how genes actually
build organisms, i.e. about embryology. The founders of the modern
synthesis treated embryology as a ‘black box’, the details
of which could be ignored for the purposes of evolutionary theory;
their focus was on the transmission of genes across generations, not
the process by which genes make organisms. This strategy was perfectly
reasonable, given how little was understood about development at the
time. In recent years, great strides have been made in molecular
developmental genetics, which has renewed hopes of integrating the
study of embryological development with evolutionary theory; hence the
emerging new discipline of ‘evolutionary developmental
biology’, or evo-devo. It is sometimes argued that evo-devo is
in tension with traditional neo-Darwinism (e.g. Amundson 2007), but it
is more plausible to view them as compatible theories with different
emphases.

In a recent book, Sean Carroll, a leading evo-devo researcher, argues
that population genetics no longer deserves pride-of-place on the
evolutionary biology curriculum. He writes: “millions of biology
students have been taught the view (from population genetics) that
‘evolution is change in gene frequencies’ … This
view forces the explanation toward mathematics and abstract
descriptions of genes, and away from butterflies and zebras, or
Australopithecines and Neanderthals” (2005 p. 294). A similar
argument has been made by Massimo Pigliucci (2008). Carroll argues
that instead of defining evolution as ‘change in gene
frequencies’, we should define it as ‘change in
development’, in recognition of the fact that most morphological
evolution is brought about through mutations that affect organismic
development. Carroll may be right that evo-devo makes for a more
accessible introduction to evolutionary biology than population
genetics, and that an exclusive focus on gene frequency dynamics is
not the best way to understand all evolutionary phenomena; but
population genetics arguably remains indispensable to a full
understanding of the evolutionary process.

Despite the criticisms levelled against it, population genetics has
had a major influence on our understanding of how evolution works. For
example, the well-known ‘gene's eye’ view of evolution,
developed by biologists such as G.C. Williams, W.D. Hamilton and
Richard Dawkins, stems directly from population-genetic reasoning;
indeed, important aspects of gene's eye thinking were already present
in Fisher's writings (Okasha 2008). Proponents of the gene's eye view
argue that genes are the real beneficiaries of the evolutionary
process; genotypes and organisms are mere temporary
manifestations. Natural selection is at root a matter of competition
between gene lineages for greater representation in the gene pool;
creating organisms with adaptive features is a ‘strategy’
that genes have devised to secure their posterity (Dawkins 1976,
1982). Gene's eye thinking has revolutionised many areas of
evolutionary biology in the last thirty years, particularly in the
field of animal behaviour; but in many ways it is simply a colourful
gloss on the conception of evolution implicit in the formalisms of
population genetics.

Population genetics raises a number of interesting philosophical
issues. One such issue concerns the concept of the gene itself. As we
have seen, population genetics came into being in the 1920s and 1930s,
long before the molecular structure of genes had been discovered. In
these pre-molecular days, the gene was a theoretical entity,
postulated in order to explain observed patterns of inheritance in
breeding experiments; what genes were made of, how they caused
phenotypic changes, and how they were transmitted from parent to
offspring were not known. Today we do know the answers to these
questions, thanks to the spectacular success of the molecular genetics
ushered in by Watson and Crick's discovery of the structure of DNA in
1953. The gene has gone from being a theoretical entity to being
something that can actually be manipulated in the laboratory.

The relationship between the gene of classical (pre-molecular)
genetics, and the gene of modern molecular genetics is a subtle and
much discussed topic (Beurton, Falk and Rheinberger (eds.) 2000,
Griffiths and Stotz 2006, Moss 2003). In molecular genetics,
‘gene’ refers, more or less, to a stretch of DNA that
codes for a particular protein—so a gene is a unit of
function. But in classical population genetics, ‘gene’
refers, more or less, to a portion of hereditary material that is
inherited intact across generations—so a gene is a unit of
transmission, not a unit of function. In many cases, the two concepts
of gene will pick out roughly the same entities—which has led
some philosophers to argue that classical genetics can be
‘reduced’ to molecular genetics (Sarkar 1998). But it is
clear that the two concepts do not have precisely the same extension;
not every molecular gene is a classical gene, nor vice-versa. Some
theorists go further than this, arguing that what molecular biology
really shows is that there are no such things as classical genes.

Whatever one's view of this debate, it is striking that virtually all
of the central concepts of population genetics were devised in the
pre-molecular era, when so little was known about what genes were; the
basic structure of population-genetic theory has changed little since
the days of Fisher, Haldane and Wright. This reflects the fact that
the empirical presuppositions of population-genetic models are really
quite slim; the basic presupposition is simply the existence of
hereditary particles which obey the Mendelian rules of transmission,
and which somehow affect the phenotype. Therefore, even without
knowing what these hereditary particles are made of, or how they exert
their phenotypic effects, the early population geneticists were able
to devise an impressive body of theory. That the theory continues to
be useful today illustrates the power of abstract models in
science.

This leads us to another facet of population genetics that has
attracted philosophers' attention: the way in which abstract models,
that involve simplifying assumptions known to be false, can illuminate
actual empirical phenomena. Idealized models of this sort play a
central role in many sciences, including physics, economics and
biology, and raise interesting methodological issues. In particular,
there is often a trade-off between realism and tractability; the more
realistic a model the more complicated it becomes, which typically
limits its usefulness and its range of applicability. This general
problem and others like it have been extensively discussed in the
philosophical literature on modelling (e.g. Godfrey-Smith 2006,
Weisberg 2006, Frigg and Hartmann 2006), and are related to population
genetics by Plutynski (2006).

It is clear that population genetics models rely on assumptions known
to be false, and are subject to the realism / tractability
trade-off. The simplest population-genetic models assume random
mating, non-overlapping generations, infinite population size, perfect
Mendelian segregation, frequency-independent genotype fitnesses, and
the absence of stochastic effects; it is very unlikely (and in the
case of the infinite population assumption, impossible) that any of
these assumptions hold true of any actual biological population. More
realistic models, that relax one of more of the above assumptions,
have been constructed, but they are invariably much harder to
analyze. It is an interesting historical question whether these
‘standard’ population-genetic assumptions were originally
made because they simplified the mathematics, or because they were
believed to be a reasonable approximation to reality, or both. This
question is taken up by Morrison (2004) in relation to Fisher's early
population-genetic work.

Another philosophical issue raised by population genetics is
reductionism. It is often argued that the population-genetic view of
evolution is inherently reductionistic, by both its critics and its
defenders. This is apparent from how population geneticists define
evolution: change in gene frequency. Implicit in this definition is
the idea that evolutionary phenomena such as speciation, adaptive
radiation, diversification, as well as phenotypic evolution, can
ultimately be reduced to gene frequency change. But do we really know
this to be true? Many biologists, particularly ‘whole
organism’ biologists, are not convinced, and thus reject both
the population-genetic definition of evolution and the primacy
traditionally accorded to population genetics within evolutionary
biology (Pigliucci 2008).

This is a large question, and is related to the issues discussed in
section 4. The question can be usefully divided into two: (i) can
microevolutionary processes explain all of evolution?; (ii) can all of
micro-evolution be reduced to population genetics?
‘Microevolution’ refers to evolutionary changes that take
place within a given population, over relatively short periods of time
(e.g. three hundred generations). These changes typically involve the
substitution of a gene for its alleles, of exactly the sort modelled
by population genetics. So over microevolutionary time-scales, we do
not typically expect to see extinction, speciation or major
morphological change — phenomena which are called
‘macroevolutionary’. Many biologists believe that
macroevolution is simply ‘microevolution writ large’, but
this view is not universal. Authors such as Gould (2002) and Eldredge
(1989), for example, have argued persuasively that macro-evolutionary
phenomena are governed by autonomous dynamics, irreducible to a
microevolutionary basis. Philosophical discussions of this issue
include Sterelny (1996), Grantham (1995) and Okasha (2006).

Setting aside the reducibility of macro to micro-evolution, there is
still the issue of whether an exclusively population-genetic approach
to the latter is satisfactory. Some reasons for doubting this have
been discussed already; they include the complexity of the
genotype-phenotype relation, the fact that population genetics treats
development as a black-box, and the idealizing assumptions that its
models rest on. Another point, not discussed above, is the fact that
population genetics models are (deliberately) silent about
the causes of the fitness differences between genotypes whose
consequences they model (Sober 1984, Glymour 2006). For example, in
the simple one-locus model of section 3.1, nothing is said about why
the three genotypes leave different numbers of successful gametes. To
fully understand evolution, the ecological factors that lead to these
fitness differences must also be understood. While this is a valid
point, the most it shows is that an exclusively population-genetic
approach cannot yield a complete understanding of the evolutionary
process. This does not really threaten the traditional view that
population genetics is fundamental to evolutionary theory.

A final suite of philosophical issues surrounding population genetics
concerns causation. Evolutionary biology is standardly thought of as a
science that yields causal explanations: it tells us the causes of
particular evolutionary phenomena (Okasha 2009). This causal dimension
to evolutionary explanations is echoed in population genetics, where
selection, mutation, migration and random drift are often described as
causes, or ‘forces’, which lead to gene frequency change
(Sober 1984). The basis for this way of speaking is obvious enough. If
the frequency of gene A in a population increases from one generation
to another, and if the population obeys the rules of Mendelian
inheritance, then as a matter of logic one of three things must have
happened: organisms bearing gene A must have outreproduced organisms
without (I); organisms bearing gene A must have migrated into the
population (II); or there must have been mutation to gene A from one
of its alleles (III). It is straightforward to verify that if none of
(I)-(III) had happened, then the frequency of gene A would have been
unchanged. Note that case (I) covers both selection and random drift,
depending on whether the A and non-A organisms reproduced
differentially because of their genotypic difference, or by
chance.

Despite this argument, a number of philosophers have objected to the
idea that evolutionary change can usefully be thought of as caused by
different factors, including natural selection (e.g. Matthen and Ariew
2009, Walsh 2007). A variety of objections to this apparently innocent
way of speaking have been levelled; some of these seem to be
objections to the metaphor of ‘evolutionary forces’ in
particular, while others turn on more general considerations to do
with causality and chance. The status of these objections is a
controversial matter; see Reisman and Forber (2005), Brandon and
Ramsey (2007) and Sarkar (2011) for critical discussion. The
‘non-causal’ (or ‘statistical’ as it is
sometimes called) view of evolution is certainly a radical one, since
the idea that natural selection, in particular, is a potential cause
of evolutionary change is virtually axiomatic in evolutionary biology,
and routinely taught to students of the subject. As Millstein (2002)
points out, if one abandons this view it becomes hard to make sense of
important episodes in the history of evolutionary biology, such as the
selectionist / neutralist controversy.

A full resolution of this issue cannot be attempted here; however, it
is worth making one observation about the idea that mutation,
selection, migration and drift should be regarded as
‘causes’ of gene frequency change. There is an important
difference between drift on the other hand and the other three factors
on the other. This is because mutation, selection and migration are
directional; they typically lead to a non-zero expected change in gene
frequencies (Rice 2004 p. 132). Random drift on the other hand is
non-directional; the expected change due to drift is by definition
zero. As Rice (2004) points out, this means that mutation, selection
and migration can each be represented by a vector field on the space
of gene frequencies; their combined effects on the overall
evolutionary change is then represented by ordinary vector
addition. But drift cannot be treated this way, for it has a magnitude
but not a direction. In so far as proponents of the
‘non-causal’ view are motivated by the oddity of regarding
drift, or chance, as a causal force, they have a point. However this
line of argument is specific to random drift; it does not generalize
to all the factors that affect gene frequency change.

A related consideration is this. Suppose that instead of selection and
drift we use the expression ‘differential reproduction’ to
cover both. This gives us three ‘factors’ that can lead to
gene-frequency change in Mendelian populations: differential
reproduction, mutation, and migration. It is straightforward to verify
that at least one of these three factors must have operated, if gene
frequencies in a population change. It seems unproblematic to regard
these three factors as causes of evolution. However, the idea that
differential reproduction can be decomposed into two
‘sub-causes’, namely natural selection and random drift,
is much more dubious. When we speak of differential reproduction as
being ‘due‘ to random drift, or chance, this is not
happily construed as a causal attribution. Rather, what we mean is
that the differential reproduction was not the result of systematic
differences in how well the genotypes were adapted to the
environment.

To conclude, it is unsurprising to find so much philosophical
discussion of population genetics given its centrality to evolutionary
biology, a science which has long attracted the attention of
philosophers. The preceding discussion has focused on the most
prominent debates surrounding population genetics in the recent
philosophical literature; but in fact population genetics is relevant,
at least indirectly, to virtually all of the topics traditionally
discussed by philosophers of evolutionary biology.

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